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Very interesting post.
As I said before, Arius didn't have a problem with “Mother of the Incarnate Word”, but he, and later Nestorius (for a different reason) had a huge problem with the notion of Mary as “Mother of God”, because he (Arius) did not believe Jesus to be truly God. He believed the Word was Incarnate in the womb of Mary, but denied that the Word was truly God from all eternity. He was “created” at some point by the Father, according to Arius.
Nestorius’ problem was a little more complicated, and has been discussed in some detail here. Just go to the Wikipedia articles (Ephesus, Nestorianism) for more detail; they're both fairly good summations of the heresy and the reason for the Council's definition in 431 AD.
The definition “Mother of God” slammed the door shut on the Trinitarian and Christological heresies which threatened the unity of the Church.
Well put. Alamo-girl, there is no simple one-worder. God is complex. We know what He is not. Mother of the Incarnate Word leaves room for the possibilities of Arius (i.e. that Christ was not God or not THE God but a “godling” or a “sub-god” or an angel or a demiurge) or Nestorius (that Christ was not wholly man and wholly God, hence God did not suffer for us) or Adoptionism (that Jesus was “possessed” by the Holy Spirit)